L'équation z^2=−1 n'admet pas de solution dans R. Nous introduisons le système des nombres complexes C : C'est un corps qui contient R et qui nous permet de résoudre cette équation. Nous introduisons d'abord différentes manières de représenter un nombre complexe. Par la suite nous discutons les solutions des équations de la forme zn=w avec n∈N∗ et w∈C . Nous terminons avec un théorème plus général sur les racines de polynômes: le théorème fondamental de l'algèbre . Ce chapitre est indépendant du reste du cours; par la suite on va presque toujours considérer les nombres réels et pas les nombres complexes.

Localization errors of the stochastic heat equation

David Jean-Michel Candil

In this thesis, we study the stochastic heat equation (SHE) on bounded domains and on the whole Euclidean space

\R^d.

We confirm the intuition that as the bounded domain increases to the whole space, both solutions become arbitrarily close to one another. Both vanishing Dirichlet and Neumann boundary conditions are considered.We first study the nonlinear SHE in any space dimension with multiplicative correlated noise and bounded initial data. We prove that the solutions to SHE on an increasing sequence of domains converge exponentially fast to the solution to SHE on

\R^d.

Uniform convergence on compact set is obtained for all

p

-moments. The conditions that need to be imposed on the noise are the same as those required to ensure existence of a random field solution. A Gronwall-type iteration argument is used together with uniform bounds on the solutions, which are surprisingly valid for the entire sequence of increasing domains.We then study SHE in space dimension

d\ge 2

with additive white noise and bounded initial data. Even though both solutions need to be considered as distributions, their difference is proved to be smooth. If fact, the order of smoothness depends only on the regularity of the boundary of the increasing sequence of domains. We prove that the Fourier transform, in the sense of distributions, of the solution to SHE on

\R^d

do not have any locally mean-square integrable representative. Therefore, convergence is studied in local versions of Sobolev spaces. Again, exponential rate is obtained.Finally, we study the Anderson model for SHE with correlated noise and initial data given by a measure. We obtain a special expression for the second moment of the difference of the solution on

\R^d

with that on a bounded domain. The contribution of the initial condition is made explicit. For example, exponentially fast convergence on compact sets is obtained for any initial condition with polynomial growth. More interestingly, from a given convergence rate, we can decide whether some initial data is admissible.

EPFL2022

Inverse Modelling and Predictive Inference in Continuum Mechanics: a Data-Driven Approach

Claire Marianne Charlotte Capelo

The explosive growth of machine learning in the age of data has led to a new probabilistic and data-driven approach to solving very different types of problems. In this paper we study the feasibility of using such data-driven algorithms to solve classic physical and mathematical problems. In particular, we try to model the solution of an inverse continuum mechanics problem in the context of linear elasticity using deep neural networks. To better address the inverse function, we start first by studying the simplest related task,consisting of a building block of the actual composite problem. By empirically proving the learnability of simpler functions, we aim to draw conclusions with respect to the initial problem.The basic inverse problem that motivates this paper is that of a 2D plate with inclusion under specific loading and boundary conditions. From measurements at static equilibrium,we wish to recover the position of the hole. Although some analytical solutions have been formulated for 3D-infinite solids - most notably Eshelby’s inclusion problems - finite problems with particular geometries, material inhomogeneities, loading and boundary conditions require the use of numerical methods which are most often efficient solutions to the forward problem, the mapping from the parameter space to the measurement/signal space, i.e. in our case computing displacements and stresses knowing the size and position of the inclusion. Using numerical data generated from the well-defined forward problem,we train a neural network to approximate the inverse function relating displacements and stresses to the position of the inclusion. The preliminary results on the 2D-finite problem are promising, but the black-box nature of neural networks is a huge issue when it comes to understanding the solution.For this reason, we study a 3D-infinite continuous isotropic medium with unique concentrated load, for which the Green’s function gives an analytical mathematical expression relating relative position of the point force and the displacements in the solid. After de-riving the expression of the inverse, namely recovering the relative position of the force from the Green’s matrix computed at a given point in the medium, we are able to study the sensitivity of the inverse function. From both the expression of the Green’s function and its inverse, we highlight what issues might arise when training neural networks to solve the inverse problem. As the Green’s function is not bijective, bijection must been forced when training for regression. Moreover, due to displacements growing to infinity as we approach the singularity at zero, the training domain must be constrained to be some distance away from the singularity. As we train a neural network to fit the inverse of the Green’s function, we show that the input parameters should include the least possible redundant information to ensure the most efficient training.We then extend our analysis to two point forces. As more loads are added, bijection is harder to enforce as permutations of forces must be taken into account and more collisions may arise, i.e. multiple specific combinations of forces might yield the same measurements.One obvious solution is to increase the number of nodes where displacements are measured to limit the possibility of collision. Through new experiments, we show again that the best training is achieved for the least possible amount of nodes, as long as the training data generated is indeed bijective. As the medium is elastic, we propose a neural network architecture that matches the composite nature of the inverse problem. We also present another formulation of the problem which is invariant to permutations of the forces,namely multilabel classification, and yields good performance in the two-load case.Finally, we study the composite inverse function for 2, 3, 4 and 5 forces. By comparing training and accuracy for different neural network architectures, we expose the model easiest to train. Moreover, the evolution of the final accuracy as more loads are added indicates that deep-neural networks (DNNs) are not well suited to fit a non-linear mapping from and to a high-dimensional space. The results are more convincing for multilabel classification.

2020

On some algebraic and extremal problems in discrete geometry

Seyed Hossein Nassajianmojarrad

In the present thesis, we delve into different extremal and algebraic problems arising from combinatorial geometry. Specifically, we consider the following problems. For any integer

n\ge 3

, we define

e(n)

to be the minimum positive integer such that any set of

e(n)

points in general position in the plane contains

n

points in convex position. In 1935, Erd\H{o}s and Szekeres proved that

e(n) \le {2n-4 \choose n-2}+1

and later in 1961, they obtained the lower bound

2^{n-2}+1 \le e(n)

, which they conjectured to be optimal. We prove that

e(n) \le {2n-5 \choose n-2}-{2n-8 \choose n-3}+2

. In a recent breakthrough, Suk proved that

e(n) \le 2^{n+O\left(n^{2/3}\log n\right)}

. We strengthen this result by extending it to pseudo-configurations and also improving the error term. Combining our results with a theorem of Dobbins et al., we significantly improve the best known upper bounds on the following two functions, introduced by Bisztriczky and Fejes T'{o}th and by Pach and T'{o}th, respectively. Let

c(n)

(and

c'(n)

) denote the smallest positive integer

N

such that any family of

N

pairwise disjoint convex bodies in general position (resp.,

N

convex bodies in general position, any pair of which share at most two boundary points) has an

n

members in convex position. We show that

c(n)\le c'(n)\le 2^{n+O\left(\sqrt{n\log n}\right)}

. Given a point set

P

in the plane, an ordinary circle for

P

is defined as a circle containing exactly three points of

P

. We prove that any set of

n

points in the plane, not all on a line or a circle, determines at least

\frac{1}{4}n^2-O(n)

ordinary circles. We determine the exact minimum number of ordinary circles for all sufficiently large

n

, and characterize all point sets that come close to this minimum. We also consider the orchard problem for circles, where we determine the maximum number of circles containing four points of a given set and describe the extremal configurations. A special case of the Schwartz-Zippel lemma states that given an algebraic curve

C\subset \mathbb{C}^2

of degree

d

and two finite sets

A,B\subset \mathbb{C}

, we have

|C\cap (A\times B)|=O_d(|A|+|B|)

. We establish a two-dimensional version of this result, and prove upper bounds on the size of the intersection

|X\cap (P\times Q)|

for a variety

X\subset \mathbb{C}^4

and finite sets

P,Q\subset \mathbb{C}^2

. A key ingredient in our proofs is a two-dimensional version of a special case of Alon's combinatorial Nullstellensatz. As corollaries, we generalize the Szemer'edi-Trotter point-line incidence theorem and several known bounds on repeated and distinct Euclidean distances. We use incidence geometry to prove some sum-product bounds over arbitrary fields. First, we give an explicit exponent and improve a recent result of Bukh and Tsimerman by proving that

\max \{ |A+A|, |f(A, A)|\}\gg |A|^{6/5}

for any small set

A\subset \mathbb{F}_p

and quadratic non-degenerate polynomial

f(x, y)\in \mathbb{F}_p[x, y]

. This generalizes the result of Roche-Newton et al. giving the best known lower bound for the term

\max \{ |A+A|, |A \cdot A |\}

. Secondly, we improve and generalize the sum-product results of Hegyv'{a}ri and Hennecart on

\max\{ |A+B|, |f(B,C)|\}

, for a specific type of function

f

. Finally, we prove that the number of distinct cubic distances generated by any small set

A\times A\subset \mathbb{F}_p^2

\Omega(|A|^{8/7})

, which improves a result of Yazici et al.

EPFL2017

Inverse Modelling and Predictive Inference in Continuum Mechanics: a Data-Driven Approach

Claire Marianne Charlotte Capelo

2020

On some algebraic and extremal problems in discrete geometry

Seyed Hossein Nassajianmojarrad

In the present thesis, we delve into different extremal and algebraic problems arising from combinatorial geometry. Specifically, we consider the following problems. For any integer

n\ge 3

, we define

e(n)

to be the minimum positive integer such that any set of

e(n)

points in general position in the plane contains

n

points in convex position. In 1935, Erd\H{o}s and Szekeres proved that

e(n) \le {2n-4 \choose n-2}+1

and later in 1961, they obtained the lower bound

2^{n-2}+1 \le e(n)

, which they conjectured to be optimal. We prove that

e(n) \le {2n-5 \choose n-2}-{2n-8 \choose n-3}+2

. In a recent breakthrough, Suk proved that

e(n) \le 2^{n+O\left(n^{2/3}\log n\right)}

c(n)

(and

c'(n)

) denote the smallest positive integer

N

such that any family of

N

pairwise disjoint convex bodies in general position (resp.,

N

convex bodies in general position, any pair of which share at most two boundary points) has an

n

members in convex position. We show that

c(n)\le c'(n)\le 2^{n+O\left(\sqrt{n\log n}\right)}

. Given a point set

P

in the plane, an ordinary circle for

P

is defined as a circle containing exactly three points of

P

. We prove that any set of

n

points in the plane, not all on a line or a circle, determines at least

\frac{1}{4}n^2-O(n)

ordinary circles. We determine the exact minimum number of ordinary circles for all sufficiently large

n

C\subset \mathbb{C}^2

of degree

d

and two finite sets

A,B\subset \mathbb{C}

, we have

|C\cap (A\times B)|=O_d(|A|+|B|)

. We establish a two-dimensional version of this result, and prove upper bounds on the size of the intersection

|X\cap (P\times Q)|

for a variety

X\subset \mathbb{C}^4

and finite sets

P,Q\subset \mathbb{C}^2

\max \{ |A+A|, |f(A, A)|\}\gg |A|^{6/5}

for any small set

A\subset \mathbb{F}_p

and quadratic non-degenerate polynomial

f(x, y)\in \mathbb{F}_p[x, y]

. This generalizes the result of Roche-Newton et al. giving the best known lower bound for the term

\max \{ |A+A|, |A \cdot A |\}

. Secondly, we improve and generalize the sum-product results of Hegyv'{a}ri and Hennecart on

\max\{ |A+B|, |f(B,C)|\}

, for a specific type of function

f

. Finally, we prove that the number of distinct cubic distances generated by any small set

A\times A\subset \mathbb{F}_p^2

\Omega(|A|^{8/7})

, which improves a result of Yazici et al.

EPFL2017

Localization errors of the stochastic heat equation

David Jean-Michel Candil

In this thesis, we study the stochastic heat equation (SHE) on bounded domains and on the whole Euclidean space

\R^d.

\R^d.

Uniform convergence on compact set is obtained for all

p

d\ge 2

\R^d

\R^d

EPFL2022

Analyse I (partie 2) : Introduction aux nombres complexes

Localization errors of the stochastic heat equation

Inverse Modelling and Predictive Inference in Continuum Mechanics: a Data-Driven Approach

On some algebraic and extremal problems in discrete geometry

Inverse Modelling and Predictive Inference in Continuum Mechanics: a Data-Driven Approach

On some algebraic and extremal problems in discrete geometry

Localization errors of the stochastic heat equation